Kernel regression with Weibull-type tails

Transcription

1 Kernel regression with Weibull-type tails Tertius De Wet, Yuri Goegebeur, Armelle Guillou, Michael Osmann To cite this version: Tertius De Wet, Yuri Goegebeur, Armelle Guillou, Michael Osmann. Kernel regression with Weibulltype tails. Annals of the Institute of Statistical Mathematics, Springer Verlag, 206, <0.007/s z>. <hal-03292> HAL Id: hal Submitted on 3 May 206 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Kernel regression with Weibull-type tails Tertius de Wet Yuri Goegebeur Armelle Guillou Michael Osmann May 8, 205 Abstract We consider the estimation of the tail coefficient of a Weibull-type distribution in the presence of random covariates. The approach followed is non-parametric and consists of locally weighted estimation in narrow neighborhoods in the covariate space. We introduce two families of estimators and study their asymptotic behavior under some conditions on the conditional response distribution, the kernel function, the density function of the independent variables, and for appropriately chosen bandwidth and threshold parameters. We also introduce a Weissman-type estimator for estimating upper extreme conditional quantiles. The finite sample behavior of the proposed estimators is examined with a simulation experiment. The practical applicability of the methodology is illustrated on a data set of sea storm measurements. Keywords: Extreme value statistics, Weibull-type distribution, regression, second order condition. AMS Subject classification: 62G05, 62G20, 62G32. Introduction The estimation of the tail index of a distribution plays a central role in the area of extreme value statistics. It is typically the first step in a practical data analysis, since such an estimate is needed in e.g. models used to estimate upper extreme quantiles. In this paper, we will consider some estimation problems within the Gumbel class, which is a rich subclass of the max-domain of attraction. Although different types of tail behavior are possible, all these distributions have in common an extreme value index γ equal to zero and thus differentiating them on the basis Department of Statistics and Actuarial Science, University of Stellenbosch, Private Bag X, Matieland 7602, South Africa tdewet@sun.ac.za. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark yuri.goegebeur@imada.sdu.dk. Institut Recherche Mathématique Avancée, UMR 750, Université de Strasbourg et CNRS, 7 rue René Descartes, Strasbourg cedex, France armelle.guillou@math.unistra.fr. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark mosma@imada.sdu.dk.

3 of this parameter alone is impossible. To solve this issue, we restrict our study to Weibull-type distributions, which have distribution functions F defined for some θ > 0 as F y = exp y /θ l y, y > 0, where l is a slowly varying function at infinity, i.e. l λy l y as y for all λ > 0. The parameter θ, called the Weibull-tail coefficient, clearly governs the tail behavior, where larger values correspond to a slower decay of F towards zero. Different values of it allow the Weibull-type distributions to cover a large part of the Gumbel class, and hence to constitute a flexible subgroup. The model finds important applications in areas such as hydrology, meteorology, environmental and actuarial science, to name but a few. For a given data set one can evaluate the adequacy of the Gumbel class by performing a test for γ = 0 versus γ 0, see e.g. Segers and Teugels The more specific Weibull-type class can be visually evaluated by an inspection of the Weibull quantile plot, which should become linear in the largest observations. Goegebeur and Guillou 200 introduced a formal goodness-of-fit test for the semi-parametric Weibull-type model based on a quantification of the linearity of the upper part of the Weibull quantile plot. In the analysis of univariate Weibull-type tails, the estimation of θ and the subsequent estimation of upper extreme quantiles assume a central position. We refer to Broniatowski 993, Beirlant et al. 995, Girard 2004, Gardes and Girard 2005, 2008a, Diebolt et al. 2008, Dierckx et al. 2009, Goegebeur et al. 200, Goegebeur and Guillou 20, and the references therein. In this paper we will consider the estimation of the Weibull-tail coefficient and of upper extreme quantiles when a random covariate X is recorded simultaneously with the variable of interest Y. Thus, we do not consider the classical experimental situation where the researcher can control X and then subsequently observe Y. Instead, X and Y are observed together, and therefore X is considered to be random. Having methodology that allows to take covariate information into account in an extreme value analysis is important as it enables one to differentiate the tail behavior in terms of one or more risk factors. Our approach to this regression on extreme values is non-parametric and based on locally weighted estimation. The regression analysis of extreme values with fixed, i.e. nonrandom, covariates has been extensively considered in the extreme value literature in a parametric, semi-parametric or nonparametric way, but mostly in the case of Pareto-type distributions. On the contrary, the development of extreme value regression methodology with random covariates is still in its infancy. Some methodology has been introduced but again focusing on the case of Pareto-type distributions. One has to wait for Daouia et al. 203 for a non-parametric estimation method that is valid in the general max-domain of attraction. In the present paper, we will develop a methodology for estimating the tail coefficient and upper extreme conditional quantiles for the class of the Weibull-type distributions, which, as mentioned above, forms a rich subset of the Gumbel class. To the best of our knowledge, this estimation problem has not been considered 2

4 in the extreme value literature. The remainder of this paper is organized as follows. In the next section we introduce two classes of estimators for the conditional Weibull-tail coefficient and study their asymptotic properties. The estimation of upper extreme conditional quantiles is considered in Section 3. In Section 4 we examine the small sample behavior of the proposed estimators by a simulation experiment, and in Section 5 we show the results for the real data example. Section 6 concludes the paper. The proofs of the main results are deferred to the appendix, while for other results we refer to the supporting information which is online available. 2 Construction and asymptotic properties Let X i, Y i, i =,..., n, be independent copies of the random vector X, Y R q R +,0, where X has a distribution with joint density function g, and the conditional survival function of Y given X = x, denoted F y; x := F y; x, is of Weibull-type, that is, for some θx > 0, F y; x = exp y /θx l y; x, y > 0, 2 where l is a slowly varying function at infinity. The model can also be defined in terms of the generalized inverse of F, denoted Q, i.e. Qp; x := inf{y : F y; x p}, 0 < p < : Qp; x = ln p θx l ln p; x, 3 where l is again a slowly varying function at infinity Bingham et al., 987, p 6. Our approach to the estimation of the Weibull-tail coefficient and of extreme conditional quantiles is based on elemental kernel statistics of the form T n t x, K := n n K hn x X i ln Y i ln ω n t +{Y i > ω n }, 4 i= where t 0, K hn x := Kx/h n /h q n, where K is a joint density function on R q, h n is a nonrandom sequence of bandwidths with h n 0 if n, {A} is the indicator function on the event A and ω n is a local non-random threshold sequence satisfying ω n if n. The parameter t is a tuning parameter, introduced to allow for more flexibility. Since θx is a tail parameter for the conditional distribution of Y given X = x, it is natural to consider statistics like 4 that involve exceedances over a high threshold, and that also occur in some neighborhood of the point of interest x R q. To obtain the limiting behaviour of 4 we need to introduce some conditions on the conditional quantile function Qp; x, the density function g of the independent variables and the kernel function K. Concerning Qp; x, we introduce the so-called second order condition on its slowly varying function l. 3

5 Assumption R There exists a constant ρx < 0 and a rate function b.; x satisfying by; x 0 as y, such that for all λ, we have lλy; x ln = by; xd ly; x ρx λ + o with D ρx λ := λ t ρx dt, and where o is uniform in λ, as y. Since ln x x for x, we have that Assumption R makes the convergence of lλy; x/ly; x to its limit, being, explicit. In fact, as shown in Geluk and de Haan 987, R implies that by; x is regularly varying with index ρx, i.e. bλy; x / by; x λ ρx as y for all λ > 0, so ρx governs the rate of this convergence. If ρx is small then the convergence is slow and the estimation of tail quantities is generally difficult. Assumption R is well accepted in the literature. It was formulated in a slightly different form in e.g. Diebolt et al and Goegebeur and Guillou 20 in the Weibull-type framework, and Gardes and Girard 2008b invoked it for tail analysis in the Fréchet max-domain of attraction. As a first step in the development of estimators for θx, we study in Lemma, given in the appendix, the local behavior of the conditional expectation of a power transformed excess mω n, t; x := E[ln Y ln ω n t +{Y > ω n } X = x]. Then, to deal with the randomness of X, we have to consider the unconditional expectation m n K, t; x := E[K hn x Xln Y ln ω n t +{Y > ω n }] which is exactly the expectation of our elemental kernel T n t. In Lemma 2, given in the appendix, we state its main asymptotic expansion under the following additional assumptions. The density function of the covariate X is assumed to satisfy a Lipschitz condition. For all x, z R q, the Euclidean distance between x and z is denoted by dx, z. Assumption G There exists c g > 0 such that gx gz c g dx, z for all x, z R q. For the weight or kernel function K we assume the following. Assumption K K is a bounded density function on R q, with support Ω included in the unit hypersphere in R q. This assumption is standard in the framework of kernel estimation. Finally, since the estimation takes place in a narrow neighborhood in the covariate space, we have to introduce a condition on the oscillation of the conditional response distribution, when considered as a function of x. This condition is formulated in terms of the conditional expectation mω n, t; x. 4

6 Assumption F The conditional excess function mω n, t; x satisfies that, for ω n, h n 0, and some T > 0, Φω n, h n ; x := sup sup mω n, t; x h n z mω n, t; x 0 if n. t [0,T ] z Ω This assumption is a smoothness condition on the underlying conditional distribution function when t = 0 and on some conditional expectations of power transformed excesses when t > 0. As illustrated in the supporting information, which is online available, this assumption is satisfied in case t = 0 by imposing some more structure on F in particular that l.; x is normalized and suitable conditions on h n and ω n. From Lemmas and 2 in the appendix, we obtain the following asymptotic expansion in case t > 0 ET t n x, K = F ω n ; x ln F ω n ; x t gxθt xγt + { + b ln F t ω n ; x; x + o + O θx ln F ω n ; x ε +Oh n + OΦω n, h n ; x}, 5 while for t = 0 ET 0 n x, K = gx F ω n ; x + Oh n + OΦω n, h n ; x. 6 The leading terms of 5 and 6 motivate our estimators for θx: ˆθ n ln ˆ F ωn ; x t T t /t n x, K x; t, K, K 2 = Γt + T n 0, t > 0, x, K 2 ˆθ 2 n x; t, K, K 2 = ln ˆ F ωn ; xt n t+ x, K t + T n t, t 0, x, K 2 where K and K 2 are kernel functions satisfying K, ˆ F ωn ; x denotes a non-parametric estimator for F ω n ; x: ˆ F ω n ; x := T 0 n x, K ĝ n x with ĝ n x := n n K hn x X i, 7 being a classical kernel density estimator for g. The estimators can also be motivated as generalisations of estimators that were initially presented in the univariate independent and identically distributed i.i.d. setting, e.g. those by Goegebeur et al. 200, to the regression case. 2 We can now establish the limiting distributions of ˆθ n x; t, K, K 2 and ˆθ n x; t, K, K 2, when appropriately normalized. These are given in Theorems and 2, respectively. 5 i=

7 Theorem Let X, Y,..., X n, Y n be a sample of independent copies of the random vector X, Y where the conditional quantile function of Y given X = x satisfies 3 with X a random vector having density function g, and assume R, G, F hold and kernel functions K and K 2 satisfying K. For all x R q where gx > 0, we have that if h n 0, ω n for n, with nh q n F ω n ; x, nh q n F ω n ; xb ln F ω n ; x; x λ, nh q+2 n F ωn ; x 0, nh q n F ωn ; x/ ln F ω n ; x ε 0 for some small ε > 0, and nh q n F ω n ; xφ 2 ω n, h n ; x 0, then nh q n F ω n ; xgx [ˆθ n x; t, K, K 2 θx] D N λ gx, θ 2 x [ Γ2t + K t 2 Γ Γ 2 t + K Γ 2 ] t + K K 2. t + Theorem 2 Under the same assumptions as in Theorem, we have nh q n F ω n ; xgx [ˆθ n 2 x; t, K, K 2 θx] D N λ gx, θ 2 xγ2t + [ 22t + K t + Γ t + K 2 2 ] 2 22t + K K 2. t + Note that the mean of the limiting distribution is not necessarily centered at zero, meaning that the estimators may show some asymptotic bias, which is common for estimators of tail parameters. Similar to the estimation of univariate Weibull-type tails see e.g. Goegebeur et al., 200, Gardes and Girard, 2008a we have that, apart from the factor gx, the mean of the asymptotic distribution only depends on λ, and not on other distributional parameters, nor on t. As expected, the asymptotic variance is increasing in θx i.e. when the tail of the conditional response distribution becomes more heavy, and also depends on the kernel functions K and K 2, as well as on the tuning parameter t. In the common situation where the kernel functions K and K 2 are assumed to be equal, the asymptotic variance expressions in Theorems and 2 simplify and are given by AVar = θ2 x K 2 2 Γ2t + t 2 Γ 2 t +, AVar 2 = θ 2 x K 2 2 Γ2t + Γ 2 t +, respectively. If we compare the two variances without the factor θ 2 x K 2 2 which is in common, we can see that no estimator performs uniformly best, although ˆθ n x; t, K, K outper- 2 forms ˆθ n x; t, K, K over a wide range of values of the parameter t. 6

8 3 Upper extreme quantile estimation In this section we study the estimation of upper extreme conditional quantiles. Let ˆQp; x denote the kernel estimator of the conditional quantile function Qp; x, i.e. where ˆ F is defined in 7. ˆQp; x := inf{y : ˆ F y; x p}, p 0,, We consider the behavior of ˆQ α n ; x for α n 0 as n. In the first instance we work under an intermediate order condition nh q nα n ln α n. Let V y; x := y /θx l y; x, so that F y; x = exp V y; x and assume that l y; x is a normalized slowly varying function see Bingham et al., 987, p 5, i.e. l y; x = cxe y εu;x u du, for y, where cx > 0 and εt; x 0 as t. In terms of V y; x we have that V y; x = cxe y εu;x u du, where εt; x /θx as t. In the sequel we will say for short that V y; x is normalized regularly varying. Note that in this case yv y; x V y; x = εy; x, a.e. Theorem 3 Let X, Y,..., X n, Y n be a sample of independent copies of the random vector X, Y where the conditional survival function of Y given X = x satisfies 2 with a normalized regularly varying function V y; x, X a random vector having density function g, and assume G, F and K hold. For all x R q where gx > 0, we have that if h n 0, α n 0 for n, with nh q nα n, nh q+2 n ln α n nh q nα n gx α n 0, and nh q nα n Φ 2 Q α n ; x + o, h n ; x 0 then [ ] ˆQ αn ; x Q α n ; x D N0, θ 2 x K 2 2. Similarly to the Weibull-tail coefficient estimators see Theorems and 2, the asymptotic variance of the extreme quantiles is increasing in θx and depends on the kernel function K. The condition nh q nα n in Theorem 3 implies that α n should be of a larger order than /nh q n, and hence, ultimately, α n > /n. As a consequence, the estimator ˆQ α n ; x cannot be used to extrapolate beyond the data range. For the purpose of estimating quantiles that are further in the upper tail than the α n quantile, we propose an estimator of Q β n ; x with τ n := ln β n / ln α n τ, if α n 0, that is in the spirit of the Weissman estimator Weissman, 978: where ˆθx is either ˆQ W β n ; x := ˆQ α n ; xτ ˆθx n, ˆθ n x; t, K, K 2 or ˆθ 2 n x; t, K, K 2. 7

9 Theorem 4 Let X, Y,..., X n, Y n be a sample of independent copies of the random vector X, Y where the conditional survival function of Y given X = x satisfies 2 with a normalized regularly varying function V y; x, X a random vector having density function g, and assume R, G, F hold and kernel functions K and K 2 satisfying K. For all x R q where gx > 0, we have that if h n 0, α n 0, τ n τ, for n, with nh q nα n and, for some small ε > 0, nh q n α n max { b ln α n ; x, h n, } ln α n ε, ΦQ α n; x + o, h n ; x 0, then nh q n α n gx ln τ n [ ] ˆQW β n ; x Q β n ; x D N0, V 2 x, where V 2 x denotes the asymptotic variance of either ˆθ n x; t, K, K 2 or ˆθ 2 n x; t, K, K 2. Remark that the extreme quantile estimator inherits the asymptotic behavior of the estimator for θx that was used. For what concerns the improvement of ˆQ W β n ; x over ˆQ α n ; x in terms of extrapolation: note that one can take e.g. β n = α τ n, for some τ >, leading to β n > /n τ, which relaxes the earlier constraint that α n > /n. By taking τ large one thus improves considerably the order of the quantile that can be estimated. Note also that the only condition that we impose on β n, namely ln β n / ln α n τ,, simplifies the interpretation in terms of extrapolation compared to Daouia et al. 203 where they impose several conditions that mix the parameters α n and β n together with several functions involving the conditional extreme value index, the conditional quantile and an auxiliary function. In that case, it is very difficult to obtain information about the possible rate of β n. 4 Simulations In this section we illustrate our methodology with a simulation experiment. For the practical implementation of our estimators we have to determine both the bandwidth parameter h n and the threshold ω n. For the latter, we take, as usual in extreme value statistics, the k + -th largest response observation in the ball Bx, h n. There is, in fact, some discrepancy between the theory, established in case of a fixed, i.e. non-random, threshold and practical use with a data-driven one. This is often observed in extreme value statistics, see for instance, Smith 987 and Davison and Smith 990, in the framework of GPD modelling of excesses, or Wang and Tsai 2009 and Goegebeur et al. 204b, for Pareto-type models. Instead of a nonrandom threshold one could have worked with a random threshold, as was done e.g. by Stupfler 203 in the general max-domain of attraction. However, this approach with a random threshold would make the theoretical derivations much more difficult. Also, in the general max-domain of attraction, Goegebeur et al. 204a observed that working with a nonrandom threshold can lead to the same asymptotic distributions as the one obtained under a random threshold as in Stupfler, 203, though the practical performance of the approach based on a random threshold is not better. One can expect that similar findings will also apply to the present paper. The 8

10 selection of h n, k is carried out using a data driven method which does not require any prior knowledge about the function θx. This approach is based on a two step procedure, where we first select the bandwidth parameter h n using a cross-validation criterion h cv := argmin h n H n where H is a grid of values for h n and n i= j= {Y i Y j } ˆF n, i Y j ; X i 2, 8 ˆF n, i y; x := n j=,j i K h n x X j {Y j y} n j=,j i K. h n x X j This criterion was introduced in Yao 999, implemented by Gannoun et al and considered in an extreme value context by Daouia et al. 20, 203. Using this bandwidth, we compute in the second step, the estimates for θx with k = 5,..., k max, where k max is an appropriately chosen value. The median of these estimates is then the estimate we use for θx. From the simulation results reported in Gardes and Girard 2008a and Goegebeur et al. 200 for the estimation of θ in the univariate framework, we deduce that the parameter k max should be chosen relatively small compared to the typical number of observations in Bx, h cv, because for Weibull-type distributions the bias of the tail estimators is in some cases important. Concerning the estimation of upper extreme conditional quantiles, we use the estimate for θx we just obtained and compute the extreme conditional quantiles for k = 5,..., k max. We use again the median of these values as our final conditional quantile estimate. We report the results for two conditional distributions of Y given X = x, which are of Weibulltype: The strict Weibull distribution Wαx, λ, F y; x = e λyαx, y > 0; αx, λ > 0, for which θx = /αx and ρx =. We consider the case λ =. The extended Weibull distribution EWαx, β Klüppelberg and Villaseñor, 993, F y; x = rye yαx, where αx > 0 and ry is a regularly varying function at infinity with index β. Here θx = /αx, by; x = β ln y/[α 2 xy] and ρx =. We choose ry = /y. In the simulation experiment we also considered some other conditional Weibull-type distributions, like the Gaussian, Gamma and perturbed Weibull distributions, but these lead to similar results and therefore, for brevity, we omit them from the paper. For the Weibull-tail coefficient θx and the covariate distribution we consider two settings: 9

11 Setting : X U0, and θx = sinπx 0 2 exp 64 x 2. 2 This function was also considered in Daouia et al. 20, though in the framework of Pareto-type distributions. The function is differentiable with several stationary points. Setting 2: X Beta.25,.5 and θx = 4 { + exp 60x /4 2 {3x 0, ]} + exp 5/2{3x, 2]} +5 6xexp 5/2{3x 2, 5/2]} {3x 5/2, 3}}, which was proposed by Gardes and Stupfler 203, also in the framework of Pareto-type modelling. This function is considered because it has a smooth but non-constant part, a constant part and also some points where it is continuous but not differentiable. For all the distributions we simulate N = 500 samples of size n = 000. As measures of efficiency we use the empirical root mean squared error and the bias, computed over points x,..., x M, regularly spread over the covariate space, as follows RMSE ξ. := M N 2, ξj x m ξx m MN m= j= Bias ξ. := M N M ξ j x m ξx m N, m= where ξ j x m is the estimator for the tail parameter ξx m, either the Weibull-tail coefficient or an upper extreme conditional quantile, from simulation run j at covariate value x m. Note that in the bias calculation we use the absolute value to avoid that positive and negative biases at different values of x cancel each other out. 2 In the estimation of θx we consider the estimators ˆθ n x; t, K, K 2 and ˆθ n x; t, K, K 2 for different values of t, and where both kernel functions are taken as the bi-quadratic kernel function Kx = 5 x 2 2 {x [, ]}. 6 The choice of the kernel function is not very crucial for the practical performance of our estimators. Note that we rerun our simulations for other choices of kernel functions, like e.g. the triweight kernel function Kx = 35 x 2 3 /32, x [, ], and the results do not change a lot. To keep the length of the paper under control, we only report the results for this bi-quadratic function. The bandwidth h n is selected using the cross validation criterion in 8 on a grid of h n {0.05, 0.075,..., 0.5}, whereafter the estimate of θx is computed as described above using k max = 25. Concerning the estimation of upper extreme conditional quantiles, we examine 0 j=

12 the behaviour of ˆQ 2 W β n ; x obtained with ˆθ n x; t, K, K and ˆθ n x; t, K, K, denoted as ˆQ W β 2 n; x and ˆQ W β n; x, respectively, and quantile levels β n = /200, /2000 and /4000. The considered quantile orders imply a rather serious extrapolation since the local sample sizes are typically much smaller than n. These quantile estimators are compared with the extreme quantile estimator from Daouia et al. 203, denoted q n RP β n x, which is developed for the full max-domain of attraction, and therefore it also applies in our context of conditional Weibull-type tails. Concerning the latter we considered constant and linear weights, leading to the estimators q n RP, β n x and q n RP,2 β n x, respectively, and both were implemented with J = 3 and J = 4, where J can be seen as the number of elemental Pickands statistics the estimator is based on. For more details we refer to the Daouia et al. 203 paper. The data driven method is the one where h and k are selected separately, which is consistent with the data driven method considered for our estimators. Note that for the estimation of the Weibull-tail coefficient θx there is to date no alternative estimation procedure available as basis for comparison. Thus, summarised, in the simulation experiment we compared the small sample behaviour of the 2 Weibull-tail coefficient estimators ˆθ n x; t, K, K and ˆθ n x; t, K, K, and that of the extreme conditional quantile estimators ˆQ W β 2 n; x, ˆQ W β n; x, q n RP, β n x and q n RP,2 β n x. The 2 estimator ˆθ n x; t, K, K outperforms ˆθ n x; t, K, K in terms of bias and RMSE, for all values of t and all distributions, and similarly, the extreme quantile estimator ˆQ W β n; x has a performance superior to that of ˆQ2 W β n; x. For what concerns the Daouia et al. 203 estimators, q n RP, β n x with J = 4 has the best performance. For brevity, in the paper we only comment on the performance of the best estimators, and draw the following conclusions In Tables and 2 we show the performance of n x; t, K, K and W β n; x, obtained with the data driven strategy on Settings and 2, respectively. The results indicate that in most cases the best estimation results are obtained for small values of the tuning parameter t. As expected, the behaviour of the extreme quantile estimator deteriorates as more extreme quantile orders are considered, though the increase in bias and RMSE is modest. Comparison of the results in Tables and 2 with Table 3 indicates that the estimator ˆQ W β n; x is better than q n RP, β n x in terms of RMSE for all values of β n and t. In terms of bias it is better for almost all β n and t. From these simulations, one can say that the proposed Weissman-type extreme conditional quantile estimator is competitive compared to the Daouia et al. 203 estimator, which is to date the only alternative. However, this result is not completely unexpected since the Daouia et al. 203 estimator is more general in that it was proposed for the general max-domain of attraction, whereas ours is specific for the class of Weibull-type distributions. 5 Case study: sea storm data To illustrate the usefulness and practical relevance of the model considered and the methodology developed in this paper we will use the sea level data presented in Chapter 6 of de Haan and ˆθ ˆQ

13 Ferreira 2006; see also de Haan and de Ronde 998 for a more elaborate discussion. This data set consists of measurements of wave height HmO and still water level SW L, both expressed in meters, recorded during 828 storm events that are relevant for the seawall called Pettemer Zeewering in the province of North Holland The Netherlands. The data have been extensively analysed in the extreme value literature. For instance, in Draisma et al. 999 the focus was on estimating the tail of the wave height distribution, ignoring the information in the variable still water level. Their analysis indicated that the HmO distribution belongs to the Gumbel class. On the other hand, de Haan and de Ronde 998 and Draisma et al took a multivariate approach and estimated the tail dependence between the two variables, i.e. when both variables are simultaneously large. Our approach is complementary to these earlier analyses in that we want to estimate tail parameters of the HmO distribution given a value for the variable SW L, where the latter is not necessarily extreme. The scatterplot of the data is shown in Figure a, and clearly indicates that higher waves tend to occur together with higher still water levels. For engineers it can be of interest to study the distribution of wave heights at a given value of still water level, e.g. to decide on the height of a sea wall or, related to this, to assess the danger of overflowing an existing sea wall. We illustrate the adequacy of the Weibull-type model 2 for the variable HmO given that SW L = 0.6, by the Weibull quantile plot of the HmO measurements with corresponding SW L in a neighborhood of SW L = 0.6, say in the interval [0.5,0.7]; see Figure b. The Weibull quantile plot clearly becomes linear in the largest observations, which supports the hypothesis of an underlying Weibull-type model for the conditional response distribution. Similar quantile plots were obtained at other positions in the covariate space. Obviously, estimators for parameters related to the conditional distribution of HmO given SW L should take into account that the available SW L measurements are in fact realisations of a random covariate. Based on these considerations we can conclude that the methodology developed in this paper is appropriate for analysing the sea storm data. As a first step we focus on the estimation of the conditional Weibull-tail coefficient of the variable HmO, considered as a function of SW L. In Figure c we show the tail coefficient estimates ˆθ n SW L; 0.2, K, K, where K is the bi-quadratic kernel function, as a function of SW L. We focus here only on the estimator ˆθ n SW L; t, K, K, since in the simulations it was found to 2 be superior to ˆθ n SW L; t, K, K in terms of bias and RMSE for all values of t. The value of t used, t = 0.2, can also be motivated from the simulation results. The tuning parameters h and k were selected according to the data driven method with h n {0.05, 0.075,..., 0.5} and k max = 25, as described in Section 4. The figure indicates that the estimates for the tail coefficient generally follow the pattern in the data in that the estimates tend to be larger at SW L positions where the extreme HmO measurements show larger spacings. To illustrate the extra flexibility of our approach, we also performed a univariate analysis of the tail of the HmO distribution, thus ignoring the information in SW L, using the mean-excess based estimator for θ proposed in Goegebeur and Guillou 20. Figure d shows these univariate estimates for the Weibull-tail coefficient of the variable HmO as a function of k. This plot suggests a stable estimate of θ for k values between 50 and 00, with median This median value is also 2

14 shown as the solid horizontal reference line in Figure c, and we also show the univariate 95% bootstrap interval based on 000 bootstrap samples. The univariate estimate thus provides some average tail coefficient estimate, and from the bootstrap interval we can conclude that it is clearly not able to provide an adequate description of the conditional tails. Next, we consider the estimation of upper extreme quantiles of the HmO distribution conditional on a given value of SW L. In Figure e we show the Weissman-type quantile estimate ˆQ W /200; SW L solid line and ˆQ W /4000; SW L dotted line as a function of SW L. This estimate represents the wave height that will be exceeded on average once in 200 and 4000 storms, respectively, for a given value of SW L. Again the estimate follows the pattern in the data, and it clearly extrapolates beyond the data range, as it should. As a simple goodness-of-fit check we also estimated a less extreme conditional quantile, namely the conditional quantile -/50, and counted the number of observations exceeding that quantile. The proportion exceeding was 0.024, which is in good agreement with the theoretical conditional exceedance probability In panel f of Figure we show the estimator q n RP, /200 SW L solid line and q n RP, /4000 SW L dotted line with J = 4 of Daouia et al This estimate also follows the pattern in the data reasonably well, though it is generally located slightly below our extreme conditional quantile estimate and also exhibits a larger variability compared to our method. Note also that due to data sparsity the Daouia et al. 203 estimate could not be computed for the very small and large values of SW L. For some values of SW L the estimate q n RP, /4000 SW L is slightly below q n RP, /200 SW L, which is due to the tuning parameter selection method. 6 Conclusion We considered the estimation of the tail coefficient and upper extreme conditional quantiles for Weibull-type distributions when there are random covariates. Two families of estimators were introduced and their asymptotic properties were derived. In extreme value statistics the bias of estimators for tail quantities is often an issue. In future research we will focus on the development of bias-corrected estimators, and on related robustness issues. Acknowledgement The authors kindly acknowledge Laurens de Haan and Ana Ferreira for providing the sea level data. This work was supported by a research grant VKR from VILLUM FONDEN and an international project for scientific cooperation PICS-646. The authors are very grateful to the editor and two anonymous referees for their helpful and constructive comments on the preliminary versions of the paper. Appendix In this appendix, we give the proofs of our main results. 3

15 First, we study the local behaviour of the conditional expectation of a power transformed excess. Lemma Case i, t = 0: mω n, 0; x = F ω n ; x. Case ii, t > 0: assume 3 and R, then for some small ε > 0 and ω n we have that { F ω n ; x mω n, t; x = ln F ω n ; x t θt xγt + + b ln F t ω n ; x; x + o θx } +O ln F ω n ; x ε. Proof of Lemma The case t = 0 is trivial so we only focus on the case where t > 0. Let p n := F ω n ; x and p n = p n / lne/ p n. Note that p n p n. From the inverse probability integral transform, and denoting V U0,, we have that mω n, t; x = E[ln QV ; x ln Qp n ; x t +] pn = ln Qv; x ln Qp n ; x t dv + p n =: I + I 2. Concerning I, from 3 and R, we obtain Inspired by the inequality [ I = p n θx ln + pn pn +b ln p n ; xd ρx + ln p n ln p n p n ln Qv; x ln Qp n ; x t dv + o] t dz. for ξ 0 and x 0, we write ξ 2 x 2 D ξ + x x 0, 9 I = p n ln p n t +θx ln p n pn pn [ ln t {θx + +b ln p n ; x ln p n ln p n [D ρx + ln p n ln p n ] + b ln p n ; x + o ] t + o} dz. ln p n 4

16 From 9 we deduce sup z [ pn pn,] and hence ln p n sup z [ pn pn,] for p n. Similarly for p n. sup z [ pn pn,] By Taylor s theorem I = =: p n ln p n t +tθ t x [ ln + ln p n ln p n pn pn { θ t x ln p n [ ln + [D ρx + pn pn ln p n t dz t ln p n +b ln p n ; xtθ t x +b ln p n ; xtθ t x [D ρx + ln p n pn pn pn pn ln p n [ ln + ] ln ln p n ln p n 2 ln p n ] = o, ln p n ] = o, ln p n ln p n t dz + o t ln p n ln p n p n ln p n t {I, + I,2 + I,3 + I,4 }. ] dz + o For I,, by straightforward calculus, we obtain ln I, = θ t t ln p n xγt + + O. ln p n Concerning I,2, use 9 to obtain I,2 tθ t x 2 ln p n pn pn } t+ dz + o = O The terms I,3 and I,4 can be analyzed in a similar way and yield I,3 = b ln p n ; xtθ t xγt + + o, b ln pn ; x I,4 = O. ln p n 5 ] dz + o ln p n ln p n.

17 Collecting the terms gives p n { I = θ t ln p n t xγt + + b ln p n ; xtθ t xγt + + o ln t } ln p n +O. ln p n For what concerns I 2 we introduce the tail quantile function U, defined as Uy; x = Q /y; x, y >, and write [ ] pn t I 2 = ln U ; x ln U ; x dv. v p n p n p n The Weibull-type distributions are a subset of the Gumbel max-domain of attraction, and hence for some positive function a.; x we have that Uy; x lim ln Uuy; x ln Uy; x = ln u, u > 0, 0 y ay; x see for instance de Haan and Ferreira 2006, p 0. Thus I 2 = a t p n ; x ln U t pn v p n ; x ln U p n ; x U p n ; x p n a p n ; x /U p n ; x Note that p n / v p n / p n, for v [ p n,. dv =: ã t ; x Ĩ 2. p n Now use Corollary B.2.0 in de Haan and Ferreira 2006 p 376, to obtain for any ε > 0 and for some c > 0 that for p n sufficiently large tε pn Ĩ 2 c dv = c p n + o, v tε ln p n tε provided tε <. p n Finally, for ã.; x we use Lemma S see supporting information according to which ã ; x Eln QV ; x ln Qp n ; x V > p n p n if p n. Now θx Eln QV ; x ln Qp n ; x V > p n = ln p n [ +θx ln b ln p n; x ln p n +b ln p n ; x ln p n 0 =: I 2, + I 2,2 + I 2,3 + I 2, ln p n dz + o [D ρx + dz ] dz ln p n 6 ] dz + o ln p n

18 With arguments similar to those used when considering I we obtain Consequently This then gives that for some small δ > 0. I 2, = θx ln p n, I 2,2 = O ln p n 2, I 2,3 = b ln p n; x + o, ln p n b ln pn ; x I 2,4 = O ln p n 2. Eln QV ; x ln Qp n ; x V > p n = O I 2 = O p n ln p n t+ δ ln p n Combining the results for I and I 2 establishes the result of Lemma. Now, to deal with the randomness of X we consider the unconditional expectation m n K, t; x and derive in the next lemma its main asymptotic expansion under further conditions. Lemma 2 Assume 3, R, G, K and F. For all x R q where gx > 0 we have that if ω n and h n 0 then,. m n K, t; x = mω n, t; xgx + Oh n + OΦω n, h n ; x. Note: i In Lemma 2, the proof of the result for the case t = 0 can also be obtained without conditions 3 and R. ii Because X i, Y i, i =,..., n, are independent and identically distributed random vectors, we have that m n K, t; x = ET n t x, K. The proof of Lemma 2 is elementary and therefore it is given in the supporting information. 2 Since the estimators ˆθ n x; t, K, K 2 and ˆθ n x; t, K, K 2 are functions of elemental kernel statistics 4, we need to derive the joint asymptotic behaviour of a vector of such statistics, t when appropriately normalized. Let T n x, K := ln F ω n ; x t T n t x, K, and for some fixed positive integer J, T n := and let Σ be a J J matrix with elements F ω n ; xgx [ T t n x, K,..., T t J n x, K J ], σ j,k := θ t j+t k x K j K k Γt j + t k +. 7

19 Lemma 3 Let X, Y,..., X n, Y n be a sample of independent copies of the random vector X, Y where the conditional quantile function of Y given X = x satisfies 3 with X a random vector having density function g, and assume R, G, F hold and kernel functions K,..., K J satisfying K. For all x R q where gx > 0, we have that if h n 0, ω n for n, with nh q n F ω n ; x, then nh q n F ω n ; xgx [T n ET n ] D N J 0, Σ. Proof of Lemma 3 We make use of the Cramér-Wold device see e.g. van der Vaart, 998, p 6, according to which it is sufficient to prove that for all ξ R J we have that Ψ n := nh q n F ω n ; xgx ξ [T n ET n ] D N0, ξ Σξ. From straightforward calculations we obtain Ψ n = n h q J n n F ξ j ln ω i= n ; xgx ω n ; x t j K j,hn x X i ln Y i ln ω n t j + {Y i > ω n } j= J E ξ j ln F ω n ; x t j K j,hn x X i ln Y i ln ω n t j + {Y i > ω n } =: j= n W i. i= Note that W,..., W n are i.i.d. random variables, and hence VarΨ n = nvarw. Now where C j,k := Cov We then have C j,k = K jk k E VarW = h q n n F ω n ; xgx J j= k= J ξ j ξ k ln F ω n ; x t j+t k C j,k, K j,hn x X ln Y ln ω n t j + {Y > ω n }, K k,hn x X ln Y ln ω n t k + {Y > ω n }. h q n [ E h q n K j K k K j x X h n [ K j,hn x X ln Y ln ω n t j + {Y > ω n } Using Lemmas and 2 we obtain that C j,k = K jk k h q n K k x X ] E h n ] ln Y ln ω n t j+t k + {Y > ω n } [ K k,hn x X ln Y ln ω n t k + {Y > ω n } F ω n ; x ln F ω n ; x t j+t k gxθ t j+t k xγt j + t k + + o, 8 ].

20 and consequently VarΨ n = ξ Σξ + o. To establish the asymptotic normality of Ψ n we verify Lyapounov s criterion for triangular arrays of random variables, see e.g. Billingsley 995, p 362. In the present context this simplifies to proving that n i= E W i 3 = ne W 3 0. We have E W 3 h q 3/2 n n F ω n ; xgx 3 J E ξ j ln F ω n ; x t j K j,hn x X ln Y ln ω n t j + {Y > ω n } j= 2 J +3E ξ j ln F ω n ; x t j K j,hn x X ln Y ln ω n t j + {Y > ω n } j= J E ξ j ln F ω n ; x t j K j,hn x X ln Y ln ω n t j + {Y > ω n } j= 3 J +4 E ξ j ln F ω n ; x t j K j,hn x X ln Y ln ω n t j + {Y > ω n }. j= Again by using Lemmas and 2 we obtain that E W 3 = O n nh q n F ω n ; x, and hence, under the conditions of the lemma ne W 3 0. i In order to obtain the limiting distribution of ˆθ n x; t, K, K 2, i =, 2, we also need a result concerning ˆ F ωn ; x as defined in 7. Daouia et al. 20 studied the asymptotic behaviour of ˆ F ω n ; x in the framework of conditional Pareto-type tails, whereas in Daouia et al. 203 its behaviour was studied for the general max-domain of attraction, though assuming that F y; x is twice differentiable. Lemma 4 below is essentially a simplified version of Proposition in Daouia et al. 203 since we only consider a single extreme level ω n, although we do not assume differentiability of F y; x nor a max-domain of attraction condition. Lemma 4 Let X, Y,..., X n, Y n be a sample of i.i.d. random vectors, and assume G, F and K hold. For all x R q where gx > 0, we have that if h n 0, ω n for n, with nh q n F ω n ; x, nh q+2 n F ωn ; x 0, and nh q n F ω n ; xφ 2 ω n, h n ; x 0 then nh q n F ω n ; xgx [ ] ˆ F ωn ; x F ω n ; x D N0, K

21 For the proof of Lemma 4 we refer to the supporting information. Proof of Theorem First we consider the estimator Write θ n x; t, K, K 2 := ln r F ω n ; x t T n t x, K n Γt + T n 0 x, K 2 = F ω n ; xgx T n 0 x, K 2 [ θ t xr n ˆθ n x; t, K, K 2 assuming that F ω n ; x is known, and introduce θ t x ln F ω n ; x t T t Γt + T 0 n x, K 2 [ ln {r F ω n ; x t T n t x, K n Γt + F ω n ; xgx T 0 n x, K 2 F ω n ; xgx E T 0 n x, K 2 F ω n ; xgx ] /t n x, K. [ E ln +r F ω n ; x t T n t x, K θ t xγt + ET n 0 x, K 2 n Γt + F ω n ; xgx =: F ω n ; xgx T n 0 x, K 2 {I 3 + I 4 + I 5 }. From Lemma 3, we have I 3 + I 4 D N 0, ln E F ] ω n ; x t T n t x, K Γt + F ω n ; xgx θ 2t x [ Γ2t + K Γ Γ 2 t + K Γ 2 ] t + K K 2. t + Concerning I 5, by using Lemmas and 2 I 5 = r n {b ln F ω n ; x; xtθ t x + o + O ln F ω n ; x ε +Oh n + OΦω n, h n ; x}, and hence under our assumptions, I 5 λ gx tθ t x. Further, from Lemma 3 we have that T 0 n x, K 2 / F ω n ; xgx = + o P. Combined, the above gives that ln r F ω n ; x t T n t x, K n Γt + T n 0 θ t D x N λ gx tθ t x, x, K 2 θ 2t x [ Γ2t + K Γ Γ 2 t + K Γ 2 ] t + K K 2. t + 20 ]}

22 A straightforward application of the δ method yields r n θ n x; t, K, K 2 θx D N λ gx, θ 2 x [ Γ2t + K t 2 Γ Γ 2 t + K Γ 2 ] t + K K 2. t + Now consider ˆθ n x; t, K, K 2 : r n ˆθ n x; t, K, K 2 θx = r n θ n x; t, K, K 2 θx ˆ F ω n ; x T +r n t /t n x, K ln F ω n ; x Γt + T n 0. x, K 2 Using Lemma 4 we have r n ln ˆ F ω n ; x = O P, F ω n ; x and by Lemmas, 2, and 3 Hence T n t /t x, K Γt + T n 0 = x, K 2 θx ln F ω n ; x + o P. r n ˆθ n x; t, K, K 2 θx = r n θ n x; t, K, K 2 θx + O P from which the result follows. Proof of Theorem 2 ln F ω n ; x The proof of this theorem follows arguments similar to those used in the proof of Theorem, and is therefore given in the supporting information. Proof of Theorem 3 The proof of the theorem is inspired by the approach taken by Wretman 978 in the univariate i.i.d. case. Let r n := ln α n nh q nα n gx/q α n ; x and r n := θx nh q nα n gx. By straightforward inversion we obtain that P r n ˆQ α n ; x Q α n ; x z = ˆ F Q α n ; x + z/ r n ; x P r n ln F Q α n ; x + z/ r n ; x r F Q α n ; x; x n ln. F Q α n ; x + z/ r n ; x, 2

23 Now let Z n := r n ln Concerning z n we have ˆ F Q α n ; x + z/ r n ; x F Q α n ; x + z/ r n ; x and z n := r n ln z n = r n V z z Q α n ; x + κ n ; x r n r n F Q α n ; x; x F Q α n ; x + z/ r n ; x. with κ n 0,, where the last step follows from the mean value theorem. Let Q α n ; x := Q α n ; x + κ n z/ r n. Note that Q α n ; x = Q α n ; x + o if α n 0, and hence, using the assumption that V is a normalized regularly varying function, we have for α n 0. Thus Q α n ; xv Q α n ; x; x V Q α n ; x; x θx V z n = r Q α n ; x; x z n + o = z + o. θx Q α n ; x r n Concerning the random term we have that Z n = n θx nh q n F Q α n ; x + z/ r n ; xgx ln where n := F Q α n ; x; x F Q α n ; x + z/ r n ; x. ˆ F Q α n ; x + z/ r n ; x F Q α n ; x + z/ r n ; x Again using 2 and by the definition of r n, one obtains that n as α n 0. Hence by Lemma 4 we have that 2 Z n D N0, θ 2 x K Denote by G n the distribution function of Z n and by G that of the limiting distribution in 3. Because of the continuity of G we have that the convergence of G n to G is uniform see e.g. Lemma 2. in van der Vaart, 998, p 2, and hence lim n G n z n = Gz. This completes the proof of Theorem 3. Proof of Theorem 4 By straightforward rearrangements we obtain nh q n α n gx ln ˆQ W β n ; x = nh q nα n gxˆθx θx ln τ n Q β n ; x nh q n α n gx + ln ˆQ α n ; x ln τ n Q α n ; x nh q n α n gx + [ln Q α n ; x ln Q β n ; x + θx ln τ n ] ln τ n =: I 6 + I 7 + I 8. 22

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26 [28] Smith, R. L., 987. Estimating tails of probability distributions. Annals of Statistics, 5, 3, [29] Stupfler, G., 203. A moment estimator for the conditional extreme-value index. Electronic Journal of Statistics, 7, [30] van der Vaart, A.W., 998. Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. [3] Wang, H., Tsai, C.L., Tail index regression. Journal of the American Statistical Association, 04, [32] Weissman, I., 978. Estimation of parameters and large quantiles based on the k largest observations. Journal of the American Statistical Association, 73, [33] Wretman, J., 978. A simple derivation of the asymptotic distribution of a sample quantile. Scandinavian Journal of Statistics, 5, [34] Yao, Q., 999. Conditional predictive regions for stochastic processes. Technical report, University of Kent at Canterbury. ˆθ Strict Weibull distribution t n x; t, K, K ˆQ W /200; x ˆQ W /2000; x ˆQ W /4000; x Bias RMSE Bias RMSE Bias RMSE Bias RMSE ˆθ Extended Weibull distribution t n x; t, K, K ˆQ W /200; x ˆQ W /2000; x ˆQ W /4000; x Bias RMSE Bias RMSE Bias RMSE Bias RMSE Table : Setting : Performance of ˆθ n x; t, K, K, ˆQ W /200; x, ˆQ W /2000; x and ˆQ W /4000; x. The results are averaged over 500 Monte Carlo simulations, with n = 000. The numbers in bold indicate the value of t with smallest bias, respectively RMSE. 25